In Section 3.4 we have encountered the addition of hours, weekdays, and months as an example for modular arithmetic. We now introduce binary operations on the sets \(\Z_n=\{0,1,2,\dots,n-1\}\) where \(n\in\N\) based on the addition and multiplication of integers. For \(a\) and \(b\) in \(\Z_n\) we consider \((a+b)\fmod n\) and \((a\cdot b)\fmod n\text{.}\) Because the remainder of division by \(n\) is always an element of \(\Z_n=\{0,1,2,\dots,n-1\}\) these yield binary operations on \(\Z_n\text{.}\)
We have already encountered operation tables for modular addition and multiplication Chapter 13. In Table 14.15 we present the operations tables for addition and multiplication modulo \(7\) side by side. Once these tables are created modular addition or multiplication can be done by table lookup.
We present examples for addition and multiplication modulo 7. Let \(a\oplus b:=(a+b)\fmod 7\) and \(a\otimes b:=(a\cdot b)\fmod 7\text{.}\) Tables for the binary operations \(\oplus\) and \(\otimes\) are given in Table 14.15.
Table14.15.Addition and multiplication tables for arithmetic modulo 7, that is, for the operations given by \(a\oplus b=(a+b)\fmod 7\) and \(a\otimes b=(a\cdot b)\fmod 7\text{.}\)
Let \(\otimes:\mathbb{Z}_{37}^\otimes\times\mathbb{Z}_{37}^\otimes\to\mathbb{Z}_{37}^\otimes\) be defined by \(a\otimes b= (a \cdot b) \bmod 37\text{.}\)
In Theorem 3.46 and Theorem 3.50 we had seen that addition and multiplication and \(\fmod\) work nicely together. These properties help make modular arithmetic easier as they help to keep the size of numbers small.
In the following two section we apply modular addition and multiplication in the definition of certain groups. We show that for any \(n\in\N\text{,}\) the set \(\Z_n\) with addition modulo \(n\) is a group and that for any prime number \(p\) the set \(\Z_p^\otimes\) with multiplication modulo \(p\) is a group.
